Seismic surveys of subsurface regions of the Earth generate seismic data sets that are analyzed to determine the geophysical properties of the subsurface regions. Inversion processes can be used to match models of the geophysical properties of the subsurface regions with the actually observed and recorded seismic data. These inversion processes include least squares migration and wave equation migration velocity analysis. These inversion processes attempt to match models of the subsurface region of interest with actual recorded seismic data from the same subsurface region through the maximization or minimization of a given cost function. These costs functions can have a connection with the estimation of the lag between seismic traces (in time, depth or any other spatial parameter). Therefore, the definition of the cost functions and accounting for this lag between the seismic traces are important in using recorded seismic data to determine the geophysical properties of a subsurface region of interest.
Regarding time lags between seismic traces, analysis of seismic surveys of a subsurface region of interest can generate two distinct datasets differing by a time difference, time lag or time shift. This time shift can be estimated by determining the location of the maximum of the cross-correlation between the two datasets over a certain window in time, possibly averaged across several traces. However, this approach does not define the time difference as a continuous function of the data samples, which is a desirable property for optimisation or when the stability of the estimate is a concern, e.g., two very similar maxima would not lead to estimates that jump from one to the other if even a tiny amount of noise is added. In addition, if the first dataset and the second dataset have a constant phase difference, e.g. two different acquisition methods having a π/2 phase difference, the maximum of the cross-correlation yields an incorrect answer as more zero-crossing points exist that indicate the time shift. Therefore, improved methods for estimating this time difference that can be used in formulating cost functions is desired.
With regard to the use of least squares migration (LSM), the classic papers are “Least-squares Crosswell Migration” by Schuster (1993) and “Least-squares Migration of Incomplete Reflection Data” by Nemeth et al. (1999). They first introduced the concept and applied it to cross-well and surface seismic data, respectively. LSM is a very computationally expensive process, which is seen in its evolution in the literature from using a Kirchhoff migration (Nemeth et al., 1999) to the more expensive but exact Reverse Time Migration (RTM) algorithm that uses the two-way wave equation (“Least-squares Migration of Multisource Data with a Deblurring Filter” by Dai et al., 2011). Some of the literature covers conventional amplitude matching LSM (“Least-squares Reverse-Time Migration” by Yao et al., 2012 and “Least-squares Reverse Time Migration: Towards True Amplitude Imaging and Improving the Resolution” by Dong et al., 2012), while others have adopted cross-correlative methods to overcome the more practical problems in the method associated with the application to real data (“A Stable and Practical Implementation of the Least-Squares Reverse Time Migration” by Zhang et al., 2013). Also, in “Practical aspects of Least-squares Reverse Time Migration” by Dong et al. (2013) using a matching filter to help with the practical problems in a LSM algorithm is addressed while the standard cost function based on the difference between the modeled data and the observed data in their LSM process is used.
Wave equation migration velocity analysis is a highly non-linear process. Hence having a more convex cost function is an important issue. This type of cost function has some connection with the estimation of lag in depth between traces. As such it makes bridges with classical migration velocity analysis where picked residual move out are used (“A Decade of Tomography” by Woodward et al., 2008), but with the benefit of not being based on picking and rays.
In general, the sinusoidal nature of seismic data means that any cost function based on the difference between observed and modeled traces can suffer from cycle skipping, causing many additional local minima to be formed in the cost function. The existence of many local minima in a cost function leads to complications in obtaining the global minima. Therefore, methods of using LSM and wave equation migration velocity analysis that avoid the problems of cycle skipping and provide a global minimum or maximum are desired. These methods could utilize cost functions that account for time lags between traces in recorded seismic data.